I am investigating how many regions can be created when n circles overlap. After I have looked at circles I will look at other shape and try to find if they have a general formula.

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Introduction

I am investigating how many regions can be created when n circles overlap. After I have looked at circles I will look at other shape and try to find if they have a general formula. When 2 Circles overlap When 3 Circles overlap When 4 circles overlap A maximum of 3 regions a maximum of 7 regions a maximum of 13 regions Can be created. Can be created. Can be created. Term U 1 2 3 4 Sequence 1 3 7 13 1st difference 2 4 6 2nd difference 2 2 2 After looking at my results so far I can see that the 1st difference is changing but the 2nd difference is constant. This tells me my equation is quadratic, and there's a formula which applies to all quadratic equations. It is: Un = an*+ bn+c First of all to find out what a is I must half the 2nd difference so A = 1, So my formula now is: Un = n*+bn+c, to find b I have to make a new sequence. ...read more.

Middle

Of 37 can be made Already I can see that there is a pattern developing that is similar to that of the circles, however the old formula does not work: U2 = 2* - 2 + 1 = 2 so it is not the same as the circles however I can work out what the formula is by using the same methods: Term U 1 2 3 4 Sequence 1 7 19 37 1st Difference 6 12 18 2nd Difference 6 6 Now from looking at this set of results I can see that this formula will be quadratic. Now to find my new formula I must do as I did first time round and half my 2nd difference, which means it is 3 so, my formula so far is now. Un = 3n - bn + c Now using the same method as before I will find out what b is Term 1 2 3 4 Sequence 1 7 19 37 3n* 3 12 27 48 New -2 -5 -8 11 Sequence 1st ...read more.

Conclusion

Un = 4n - 4n + c And I can also see that c will be +1 so my formula is now Un = 4n - 4n + 1 This is my final formula for squares I will now test it for 5 squares U5 = 4 x 5* - 4 x 5 +1 U5 = 100 - 15 + 1 = 81 This is correct as it follows the pattern so that is my final formula for squares. I do not need to draw 5 squares as I know this is right as it worked for the circles Conclusion After looking over and analysing my results I can see that a & b is always the same and is always equal to the amount of sides the shape has for example triangle 3n and square 4n the circles did not show this as circles have no sides I have also found the general formula for finding out how many regions are in any shape is: Un = an*+ bn+c Created by Anthony Godsell ...read more.

Testing my Theory The maximum product that can be retrieved by the number 8 is 16. You get this by halving 8, which is 4 and then multiplying by itself, which gives you, 16. I will test this now. (1,7)= 8 à 1+7 à 1x7=7 (2,6)= 8 à 2+6 à

Diagram 3C: (all 3 lines cross each other, regions are depicted with numbers, crossover points are high-lighted with red circles) Diagram 3 C represents the most regions and crossover points possible for three straight lines. Therefore I can deduce that in order to create the maximum number of crossover points

The nth term for this is n x 8. For the fifth column the numbers go up in ones because an extra symbol is added to the middle when the cuboid size increases. The nth term is (n+1) x4. Here are my predictions for other cubes with different lengths.

even powers of are natural powers of 2, it follows that this power series produces an even number, and a half of an even number is a whole number, therefore is a formula which produces whole numbers. If instead, one were to find the difference between and , one should

The Link The series above is a geometric series. I know this because the difference is different each time. The general way to write a geometric series is: General: a + ar + ar2 + ar3 +... arn-1 The terms: a is the starting number in the sequence. I will use a 6 tiled sequence so my starting number